Model-based fault detection in a motor drive

ABSTRACT

A method for detecting an open winding in a motor. The method employs passive monitoring of the voltage, current and speed of the motor. A residue voltage is calculated that equals the difference between an idealized set of voltage drops across the motor load elements and the actual voltage drops. When the magnitude of the residue voltage equals or exceeds a threshold, an open winding condition may be declared and appropriate action may be taken.

TECHNICAL FIELD

The present invention pertains to a method for detecting a faultcondition in a motor, such as open motor winding, using residue voltagedifferences and other passive measures.

BACKGROUND OF THE INVENTION

Dynamically stabilized transporters refer to personal vehicles having amotion control system that actively maintains the stability of thetransporter while the transporter is operating. The motion controlsystem maintains the stability of the transporter by continuouslysensing the orientation of the transporter, determining the correctiveaction to maintain stability, and commanding the wheel motors to makethe corrective action. If the transporter loses the ability to maintainstability, the rider may experience discomfort at the sudden loss ofbalance. The risk of such discomfort may be reduced if redundantcomponents are provided in the transporter drive train. For example,providing dual-stators in the motor driving the transporter's groundcontacting elements (e.g., wheels) reduces likelihood of loss ofbalance. When redundant components are provided, a method for detectingfailure of a redundant component is desirable so that a failed componentmay be replaced before a double failure occurs.

Active detection of an open motor winding, namely a periodic attempt toforce current into the motor to distinguish between a normal motor andone with an open winding, may not be feasible without requiring themotion control system to give up some control over the motor's torqueproduction. A method for passively monitoring motor winding circuits todetermine open circuit conditions would advantageously allow such opencircuits to be detected without disturbing motor operation.

SUMMARY OF THE INVENTION

In an embodiment of the present invention, a method is provided fordetecting an open winding in a motor. The method employs passivemonitoring of the voltage, current and speed of the motor. A residuevoltage is calculated that equals the difference between an idealizedset of voltage drops across the motor load elements and the actualvoltage drops. When the magnitude of the residue voltage equals orexceeds a threshold, an open winding condition may be declared andappropriate action may be taken.

In another embodiment of the invention, a dual-stator redundant motor isprovided. The method employs passive monitoring of the voltage, currentand speed of the motor for each of the dual-stators of the motor.Residue voltages are calculated for each stator that measure thedifference between an idealized set of voltage drops across therespective motor load elements and the actual voltage drops. When themagnitude of the difference of these two residue voltages equals orexceeds a threshold, an open winding condition may be declared andappropriate action may be taken. In a further embodiment of theinvention, an open winding may be declared when either the condition onthe difference of the residues is met or when the magnitude of either ofthe residues of the individual-stators equals or exceeds a threshold. Ineach of these embodiments, an open motor winding or other causes of anopen winding circuit, such as an open relay, a broken wire or an openfuse link may be detected.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing features of the invention will be more readily understoodby reference to the following detailed description, taken with referenceto the accompanying drawings, in which:

FIG. 1 is a side view of a personal vehicle lacking a stable staticposition, for supporting or conveying a subject who remains in astanding position thereon;

FIG. 2 shows a block diagram of the system architecture of an embodimentof the present invention;

FIG. 3 is a block diagram for an algorithm for detecting openingwindings in a dual-stator redundant motor according to an embodiment ofthe invention;

FIG. 4 is a block diagram of an exemplary reference transform accordingto an embodiment of the invention; and

FIGS. 5A-5D illustrate filter lag compensation.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

The subject matter of this application is related to U.S. Pat. Nos.5,701,965; 5,971,091; 5,791,425; 6,302,230 and U.S. patent applicationSer. No. 09/687,789, attorney's docket 1062/C40, “TransporterImprovements,” filed Oct. 13, 2000, which are all incorporated herein byreference in their entirety.

In an embodiment of the present invention, a method is provided fordetecting an open winding in a motor. The method employs passivemonitoring of the voltage, current and speed (or equivalently rotationalfrequency) of the motor. A residue voltage is calculated that equals thedifference between an idealized set of voltage drops across the motorload elements and the actual voltage drops. When the magnitude of theresidue voltage equals or exceeds a threshold, an open winding conditionmay be declared and appropriate action may be taken.

In another embodiment of the invention, a dual-stator redundant motor isprovided. The method employs passive monitoring of the voltage, currentand speed of the motor for each of the dual-stators of the motor.Residue voltages are calculated for each stator that measure thedifference between an idealized set of voltage drops across therespective motor load elements and the actual voltage drops. When themagnitude of the difference of these two residue voltages equals orexceeds a threshold, an open winding may be declared and appropriateaction may be taken. In a further embodiment of the invention, an openwinding condition may be declared when either the condition on thedifference of the residues is met or when the magnitude of either of theresidues of the individual-stators equals or exceeds a threshold. Ineach of these embodiments, an open motor winding or another cause of anopen circuit, such as an open relay, may be detected.

Embodiments of the present invention will be described for a dynamicallybalancing transporter. These embodiments are presented by way ofillustration and not for limiting the scope of the invention asdescribed by the appended claims. As those skilled in the art willrecognize, the present invention may be used in any device wheredetection of an open winding circuit is desired.

Dynamically-Balancing Transporter

A personal transporter may be said to act as dynamically ‘balancing’ ifit is capable of operation on one or more wheels but would be unable tostand on the wheels but for operation of a control loop governingoperation of the wheels. A balancing personal transporter lacks staticstability but is dynamically balanced.

An embodiment of a balancing personal transporter is depicted in FIG. 1and designated generally by numeral 10. User 8 is shown in FIG. 1,standing on platform (or ‘base’) 12 of ground-contacting module 26.Wheels 20 and 21 are shown as coaxial about the Y axis. A handlebar 14may be provided on stalk 16 for gripping by the user.

Referring now to FIG. 2, a block diagram is shown of a systemarchitecture for an embodiment of the present invention. This blockdiagram shows the architecture for controlling and driving one wheel 20of transporter 10. An analogous block diagram applies to controlling anddriving the other wheel 21 of the transporter. A motor 120 drives wheel20 of the transporter. The motor 120 is preferably DC brushless, but maybe either AC or DC motors and either brushed or brushless. The motor 120is energized by a redundant set of windings 121, 122. Both windings arecapable of energizing the motor either independently or simultaneously.Motor 120 has a sensor 123 that measures the position or angularvelocity of the motor shaft. Conversion of a signal representinginstantaneous shaft velocity to or from a signal representing positionis accomplished by integrating or differentiating the signal,respectively.

Processor 135 monitors various parameters of each winding 121, 122 viasensors 137, 147 that monitor at least the voltage and current for eachwinding. Processor 135 controls the voltage or current applied to eachwinding via the A winding motor drive 133 and the B winding motor drive143. The A winding motor drive 133 derives power from the A power supply131 and the B winding motor drive derives power from the B power supply141.

Open Winding Detection

A method is described below for detecting an open winding in a motor 120according to an embodiment of the invention. By way of example, but notfor limitation, this method may be performed using a computer processor,such as processor 135 in the architecture described above.

This method assumes that:

-   -   1. The electrical relationship between motor terminal voltage,        current, and motor speed is approximately known under normal        circumstances, and changes in the case of an open winding.    -   2. In a system containing a motor with a dual—redundant winding,        at most one of the two stators will contain an open winding.        Further, the electrical parameters of the motor (e.g., back-EMF        constant Ke, resistance R, and inductance L) will be reasonably        matched between the two half-motors.

A quantity, which will be called the “residue voltage” in thisspecification and in any appended claims, can be calculated bysubtracting estimated values of the components that make up motorvoltage, i.e., back-EMF and IR voltage drops, from the measured value ofthe motor voltage itself. The residue voltage effectively compares theactual value of motor voltage to its expected value during normaloperation. A large residue voltage may indicate the presence of a faultcondition, which could be due to an open winding or an open relay oranother open component in the winding circuit. The term “open windingcondition” in this specification and in any appended claims will beunderstood to mean an open winding or another cause of an open circuitin the motor. Further, “open winding condition” will also include:improper measurement of motor voltage, current and/or speed or amismatch between motor parameters and their estimated values. Ideally,when there is no open winding condition, this residue voltage shouldalways be zero. Some assumptions about the motor resistance and back EMFconstant must be made to calculate these voltage drops and precisemeasurements of motor voltage, current and frequency must be obtained.In reality, these ideal conditions do not occur so errors present inthese numbers can produce significant residue levels. An analysis ofthese errors is presented in Appendix A. Further, in this embodiment ofthe invention, the measurements of motor speed, current and voltage areall filtered at different frequencies, contributing to a non-zeroresidue voltage even with no open winding condition. Therefore, dataacquisition filter lags are accounted for in the residue calculation, asdescribed in Appendix B. Under these assumptions, it is possible toobserve an open winding passively (in other words without requiring anychanges to motor commands), when the torque commanded is sufficientlylarge to produce an observable effect.

For a DC motor, the following equation holds:$V = {{K_{e} \cdot \omega_{m}} + {I \cdot R} + {L\frac{\mathbb{d}I}{\mathbb{d}t}}}$

-   -   where:    -   V is the voltage across the motor,    -   K_(e) is the back-EMF constant of the motor,    -   ω_(m) is the mechanical speed of the motor,    -   I is the current through the motor,    -   R is the motor winding resistance, and    -   L is the motor winding inductance.

Under steady-state operation, $L\frac{\mathbb{d}I}{\mathbb{d}t}$is approximately zero and a residue, r, can be calculated.r=V−K·ω _(m) −I·R

where V, I, and ω_(m) are measured quantities and K_(e) and R areestimates of motor parameters.

If r is approximately zero, then the relationship between measured motorvoltage, current and speed matches what is expected and the motor can beassumed to be operating normally.

If r is non-zero, it may indicate that a fault (an open windingcondition) has occurred. Such faults may include:

-   -   an open winding or broken wire has occurred;    -   voltage, current and/or speed may be measured improperly; or    -   motor parameters K_(e) and R do not match their estimated        values.

Similar equations hold true for a three-phase permanent magnetsynchronous motor (“PMSM”).

In the synchronous (rotor) reference frame of a PMSM, the followingequations hold when no open winding condition is present:$\begin{matrix}{V_{qLN} = {{K_{eLN} \cdot \omega_{m}} + {I_{q} \cdot R_{LN}} + {I_{d} \cdot \omega_{e} \cdot L_{LN}} + {L_{LN}\frac{\mathbb{d}{Iq}}{\mathbb{d}t}}}} \\{V_{dLN} = {{I_{d} \cdot R_{LN}} - {I_{q} \cdot \omega_{e} \cdot L_{LN}} + {L_{LN}\frac{\mathbb{d}I_{d}}{\mathbb{d}t}}}}\end{matrix}$where:

-   the subscript “LN” denotes line-neutral quantities;-   “d” refers to the direct-axis of the synchronous reference frame,    where currents are non-torque producing and voltages are out of    phase with back-EMF;-   “q” refers to the quadrature-axis of the synchronous reference    frame, where currents are torque producing and voltages are in-phase    with back-EMF;-   K_(eLN) is the motor's back-EMF constant;-   I_(q) and I_(d) are the synchronous-frame components of motor    current;-   V_(qLN) and V_(dLN) are the synchronous-frame components of    line-neutral motor voltage;-   R_(LN) is the line-neutral resistance;-   L_(LN) is the line-neutral inductance;-   ω_(m) is the mechanical speed of the motor;-   ω_(e) is the electrical frequency, equal to p/2 times ω_(m), where p    is the number of motor poles;-   d( )/dt is differentiation with respect to time.

For the steady state case where$\frac{\mathbb{d}I_{q}}{\mathbb{d}t}\quad{and}\quad\frac{\mathbb{d}I_{d}}{\mathbb{d}t}$

-   are zero, a residue voltage, r_(q), may be calculated where:    r _(q) =V _(qLN) −I _(q) ·R _(LN) −K _(eLN)·ω_(m) ·I _(d)·ω_(m) ·L    _(LN),    where V_(qLN), I_(q), I_(d) and ω_(m) are derived from measurements,    while R_(LN) and K_(eLN) are estimated. r_(q) has units of volts,    line-to-neutral. Note that if I_(d) is controlled to zero, then the    last term in the preceding equation can be ignored.

When no open winding condition is present, r_(q) is approximately zero.

When there is either an entire set of open windings or a single openwinding, the residue voltage, r_(q) tends to be non-zero for either oftwo cases:

-   -   1. When the motor is commanded in a voltage mode and when the        commanded V_(qLN) differs from the internal back-EMF,        K_(eLN)·ω_(m), i.e. current would flow in a normal motor but        cannot due to the open winding; or    -   2. when the motor is commanded in a current mode and the current        commanded is non zero.        In either of these cases, the residue of a system with an open        winding begins to diverge from zero (the expected residue for a        normal system), and the open winding can be detected. Current        mode motor commands tend to produce larger residues because the        motor drive is actively trying to force current through the        motor, and in the case of an open winding, V_(qLN) becomes very        large (at all times in the case of an open winding set, and at        various times depending on the speed and electrical angle for a        single open winding in a winding set). In voltage mode V_(qLN)        becomes only as large as its command (with some torque ripple in        a motor with a single open winding, because this is an        unbalanced load).

In practice, the residue voltage can be compensated for differences inthe time delays that are introduced by analog low-pass-filters on thevoltage, current, and speed sensor inputs, by calculating the residue inthe following manner:r _(q) =V _(qLN) −I _(q) ·R _(LN)+(K _(d) ·V _(dLN) −K _(eLN))·ω_(m)where $K_{d} \approx {\frac{p}{2} \cdot {\Delta\tau}}$p=number of motor poles

-   -   Δτ=time delay, between position and voltage sense.        Such compensation greatly improves the accuracy of detection.        The derivation for this equation is discussed in Appendix B.

Calculation of the residue voltage, r_(q), provides a test that can beused in a motor drive, regardless of whether it is in a single-statormotor or a dual-stator redundant motor, namely:B _(ow)=(|LPF(r _(q))|>R _(THRESH))

where B_(ow) is a Boolean value that represents whether an open windingcondition has been detected and “LPF” means that the value of r_(q) hasbeen filtered with a low-pass filter. The low-pass filter's cutofffrequency should be a compromise between rejecting high-frequency errorsand a sufficiently rapid detection. A value of ≈1.5 Hz has been usedadvantageously in a dynamically balancing transporter.

In a system with a dual-stator redundant motor, two residues r_(qA) andr_(qB) may be calculated. A second test that can be then applied is:B _(ow) _(—) _(AB)=(|LPF(r _(qA))−LPF(r _(qB))|>R _(THRESH) _(—) _(AB)),where B_(ow) _(—) _(AB) is also a Boolean value that represents whetheran open winding condition has been detected.

The system can use both bits, namely, if(B_(ow)=TRUE) or (B_(ow) _(—) _(AB)=TRUE)then an open winding condition is detected: take appropriate action. Ifboth halves of the redundant motor are driven with similar voltagecommands, then B_(ow) _(—) _(AB) is more sensitive in detecting openwindings than B_(ow) alone, because some of the errors listed inAppendix A cancel out partially or completely.

Note, however, that calculating B_(ow) _(—) _(AB) requires somecommunication between the two motor drives controlling current into thetwo stators, A and B. Further, r_(qA) and r_(qB) used in the aboveequation should correspond to the same instant in time, so that if sideA gets r_(qB) with a delay, the same delay should be incorporated in itsown residue, r_(qA), before subtracting the two residues.

A block diagram for the algorithm for this embodiment of the inventionis depicted in FIG. 3. Typical values of update rate are shown for thevarious blocks. First, the residue voltage, r_(q), is calculated 310.This voltage is run through a low-pass filter 320 and then sampled 330at a 100 Hz rate. The filtered value of r_(q) 340 is passed as an inputto a corresponding algorithm for the other stator. The difference intime between the values of r_(q) for each side is compensated 350 and avoltage difference is formed 360. Finally, the voltage difference iscompared to the threshold 370.

Motor Measurements

Voltages V_(q) and V_(d) and currents I_(q) and I_(d) are used insynchronous-frame control algorithms for three-phase motor drives: phasevoltages and currents, which oscillate at the motor's electricalfrequency, are changed to DC or slowly-varying quantities which can bemore easily controlled with zero steady-state error.

A block diagram of an exemplary reference transform is shown in FIG. 4

Referring to FIG. 4, where V_(a), V_(b), V_(c) are measured motor phasevoltages and θ_(e) is an electrical angle derived from a position sensor(e.g. resolver, encoder, etc.), the “abc/xy” and “xy/dq” blocks functionas follows:V _(x)=⅔V _(a)−⅓V _(b)−⅓V _(c;)V _(y)=1/{square root}{square root over (3)}(V _(b) −V _(C));V _(d) =V _(x) sin(θ_(e))−V _(y) cos(θ_(e)); andV _(q) =V _(x) cos(θ_(e))+V _(y) sin(θ_(e)).

Equivalent formulations of these equations will be apparent to thoseskilled in the art. All formulations have the property that ifV _(a) =A cos(θ_(e))+B sin(θ_(e))+C;V _(b) =A cos(θ_(e)−120°)+B sin(θ_(e)−120°)+C; andV _(c) =A cos(θ_(e)−240°)+B sin(θ_(e) −120°)+ C.

then the transformation yields V_(q)=A and V_(d)=B (or V_(q)=−B andV_(d)=A in some formulations). Thus, a three-phase set of oscillatingwaveforms is transformed into a pair of DC values which are sufficientto describe the magnitude and phase of the original signals. The sametransform equations may be used to calculate I_(d) and I_(q) from phasecurrents I_(a), I_(b), and I_(c).

It should be understood that measuring voltages V_(q) and V_(d) andcurrents I_(q) and I_(d) implies deriving them from measured phasevoltages and currents and measured motor positions. Likewise, in thisspecification and in any appended claims, unless context requiresotherwise, “measuring a speed” includes direct speed measurement ortaking a series of position measurements with associated times and thencalculating a speed.

Algorithm Initialization

The residue voltage filters need to be zeroed after a sufficientinterval has elapsed after transporter startup. For a dual-stator motor,the algorithm results may be ignored until the remote data communicationbusses described in connection with FIG. 2 have been synchronized. Theresponse to the algorithm may be disabled for a fixed delay, such as 250milliseconds, after this initialization as added insurance against afalse positive at start-up. The primary issue is that until the filterson both sides are zeroed, the delta residue can be quite large,especially if one side has been zeroed and the other has not yet. Thetransport delay between the two sides can further complicate matters.The one-time 250 ms suppression of the response to the algorithm atstartup more than adequately addresses this concern.

The described embodiments of the invention are intended to be merelyexemplary and numerous variations and modifications will be apparent tothose skilled in the art. All such variations and modifications areintended to be within the scope of the present invention as defined inthe appended claims.

Appendix A: Sources of Error in Residue-Based Open-Winding Detection

Residue voltage equation:

-   -   theoretical: r_(q)=V_(qLN)−K_(eLN)ω_(m)−I_(q) R_(LN)    -   computed: r_(q)={circumflex over ({circumflex over        (V)})}_(qLN)−{circumflex over (K)}_(eLN){circumflex over        (ω)}_(m)−Î_(q){circumflex over (R)}_(LN), {circumflex over        ({circumflex over (V)})}_(qLN)={circumflex over        (V)}_(qLN)+{circumflex over (K)}_(d){circumflex over        (V)}_(dLN){circumflex over (ω)}_(m)        Terminal voltage equations for 3-phase permanent magnet        synchronous motor: $\begin{matrix}        {V_{qLN} = {e_{q} + {I_{q}R_{LN}} + {I_{d}\omega_{e}L_{LN}} + {L_{LN}\frac{\mathbb{d}I_{q}}{\mathbb{d}t}}}} \\        {V_{dLN} = {{e_{d} + {I_{d}R_{LN}}} = {{l_{q}\omega_{e}L_{LN}} + {L_{LN}\frac{\mathbb{d}l_{d}}{\mathbb{d}t}}}}} \\        {e_{q} = {K_{eLN}\omega_{m}\cos{\overset{\sim}{\quad\theta}}_{e}}} \\        {e_{d} = {K_{eLN}\omega_{m}\sin\quad{\overset{\sim}{\theta}}_{e}}}        \end{matrix}$

({tilde over (θ)}_(e) is error in estimate of electrical angle, normally0) Significance (Δ: is this term reduced when comparing halves of aredundant motor? Approx. (small- typical Source of error signal) errorin r_(q) qualitative quantitative Δ ${\overset{\sim}{K}}_{eLN}$${- {\overset{\sim}{K}}_{eLN}}\omega_{m}$ Large 7% V₀ yes$\begin{matrix}{{Motor}\quad{parameter}\quad{tolerance}} \\{V_{0} = {\frac{V_{bus}}{\sqrt{3}} \approx {{full}\quad{scale}\quad{line}\text{-}}}} \\{{neutral}\quad{{voltage}.}}\end{matrix}\quad$ $\begin{matrix}{{\overset{\sim}{R}}_{LN} = {R_{LN} - {\hat{R}}_{LN}}} \\\left( {{actual} - {estimate}} \right)\end{matrix}\quad$ ${- I_{q}}{\overset{\sim}{R}}_{LN}$ Possibly large10% I_(q)R_(LN) yes (I_(qdm) ≈ 0¹) Motor parameter tolerance${\overset{\sim}{\omega}}_{m}$$\left( {{K_{d}V_{dLN}} - K_{eLN}} \right)\overset{\sim}{\omega}$negligible Speed estimate error LPF makes only the DC component of{tilde over (ω)}_(m)significant; for any sensors used for commutation,this is 0. ${\overset{\sim}{I}}_{q}$ ${- {\overset{\sim}{I}}_{q}}R_{LN}$small to moderate 2-5% I_(q)R_(LN) Current sensor tolerance${\overset{\sim}{V}}_{qLN}$ ${\overset{\sim}{V}}_{qLN}$ Moderate 2% V₀Voltage sensor tolerance (resistor divider) I_(d) ≠ 0 I_(d)ω₃L_(LN)negligible Current controller error: I_(d) is controlled to 0 with a PIloop, so when r_(q) is LPF, this term disappears¹Assuming differential mode I_(q) between halves of a redundant motor issmall, this term is reduced.

Significance (Δ: is this term reduced when comparing halves of aredundant motor? Source Approx. (small- typical of error signal) errorin r_(q) qualitative quantitative Δ$\frac{\mathbb{d}l_{q}}{\mathbb{d}t} \neq 0$$L_{LN}\frac{\mathbb{d}l_{q}}{\mathbb{d}t}$ very small$2L_{LN}\frac{l_{qlim}}{\tau_{LPF}}$ yes (I_(qdm) ≈ 0) Changing currentcommand: LPF makes this term small except for slow ramp rates from fullregen to full motoring current (or vice-versa)${\overset{\sim}{\theta}}_{e}$$K_{eLN}{\omega_{m}\left( {1 - {\cos\quad{\overset{\sim}{\theta}}_{e}}} \right)}$small to moderate $\begin{matrix}\left. {15{^\circ}}\rightarrow{3.4\%\quad V_{0}} \right. \\\left. {10{^\circ}}\rightarrow{1.5\%\quad V_{0}} \right. \\{\quad\left. {5{^\circ}}\rightarrow{0.4\%\quad V_{0}} \right.}\end{matrix}\quad$ Error in electrical angle estimate: LPF makes only DCcomponent of {tilde over (θ)}_(e)significant ${\overset{\sim}{K}}_{d}$${\overset{\sim}{K}}_{d}V_{dLN}\omega_{m}$ Small 10% K_(d)V_(dLN)ω_(m)Error in phase lag compensation due to filter time constant uncertainty.(R and C tolerances) K_(d)ω_(m) is a filter phase lag and should beunder 0.25 radians, which would make this term about 2.5% V_(dLN), whichis significant only at high power levelsExample system: max I_(q)=35 A, V₀=42V (72 Vbus), R_(LN)=0.14 ohm,L_(LN)=0.44 mH, τ_(LPF)=0.1 s, K_(d)ω_(m)≦0.2 rad ω_(e)≦1880 rad/s (300Hz) $\begin{matrix}{{{\overset{\sim}{R}}_{LN}:{{largest}\quad I_{q}R_{LN}}} = {{4.9\quad V} = {12\%\quad V_{0}\quad\left( {{so}\quad{in}\quad{this}\quad{case}\quad{\overset{\sim}{R}}_{LN}} \right.}}} \\\left. \left. {{and}\quad{\overset{\sim}{I}}_{q}\quad{terms}\quad{are}\quad{small}} \right)\rightarrow{{effect}\quad{of}\quad{\overset{\sim}{R}}_{LN}\quad{is}\quad{about}} \right. \\{1.2\%\quad V_{0}\quad{worst}\quad{case}} \\{{{\frac{\mathbb{d}I_{q}}{\mathbb{d}t} \neq 0}:{{largest}\quad{LPF}\left( {L_{LN}\frac{\mathbb{d}I_{q}}{\mathbb{d}t}} \right)}} = {{2L_{LN}\frac{I_{qlim}}{\tau_{LPF}}} = {{0.31V} = {0.7\%\quad V_{0}}}}} \\{{\overset{\sim}{K}}_{d}:{V_{dLN} \approx {{- I_{q}}\omega_{e}L_{LN}} \leq {29\quad V}}} \\{{{10\%\quad K_{d}V_{dLN}\omega_{m}} \leq {0.1 \times 0.2 \times 29V}} = {{0.58V} = {1.4\%\quad V_{0}}}}\end{matrix}$So the largest sources of error in LPF(r_(q)) are probably {tilde over(K)}_(eLN) (7% V₀ at high speeds) and {tilde over (V)}_(qLN) (2%), with{tilde over (R)}_(LN), {tilde over (K)}_(d), and {tilde over (θ)}_(e) inthe 1% range and everything else under 1%. Comparing r_(q) betweenredundant halves should greatly reduce the {tilde over (K)}_(eLN) term.Appendix B: Filter Lag CompensationIf we are sensing motor phase voltages using the method depicted in FIG.5A and V_(a), V_(b), V_(c) are sensed after passing through a low-passfilter,${{H(s)} = \frac{1}{1 + {\tau\quad{s\left( {{higher}\quad{order}\quad{terms}} \right)}}}},$this will cause an error in the derived values V_(d), V_(q). as shown inFIG. 5B. The xy/abc transformations are linear and this model can besimplified to the algorithm illustrated in FIG. 5C.

This can be further simplified to the algorithm shown in FIG. 5D, where$\omega_{e} = {\frac{{\mathbb{d}\theta}\quad e}{\mathbb{d}t}.}$So that a filter acting in the stationary frame is equivalent to thesame filter, frequency shifted by the electrical frequency of the motor,in the synchronous frame.

One effect of this, is that at DC in the synchronous frame,$\begin{matrix}{{{\hat{V}}_{q} - {j{\hat{V}}_{d}}} = {{H\left( {j\quad\omega_{e}} \right)} \cdot \left( {V_{q} - {jV}_{d}} \right)}} \\{\approx {\frac{1}{1 + {\tau \cdot \left( {j\quad\omega_{e}} \right)}} \cdot \left( {V_{q} - {jV}_{d}} \right)}}\end{matrix}$(we can drop higher order terms if (ω_(e))τ<<1)

This attenuates the V_(dq) vector slightly and rotates it slightly. Wecan compensate for this effect:{circumflex over ({circumflex over (V)})} _(q) −j{circumflex over({circumflex over (V)})} _(d)=({circumflex over (V)} _(q) −j{circumflexover (V)} _(d))·(1+jω _(e)τ)≈V _(q) −jV _(d){circumflex over ({circumflex over (V)})} _(q) ={circumflex over (V)}_(q) +{circumflex over (V)} _(d)·ω_(e) τ={circumflex over (V)} _(q)+{circumflex over (V)} _(d) ·K _(d)ω_(m){circumflex over ({circumflex over (V)})} _(d) ={circumflex over (V)}_(d) −{circumflex over (V)} _(q)·ω_(e) τ={circumflex over (V)} _(d)−{circumflex over (V)} _(q) ·K _(d)ω_(m)where $K_{d} = {\frac{P}{2} \cdot \tau}$P=# of motor poles and

-   where {circumflex over ({circumflex over (V)})}_(q),{circumflex over    ({circumflex over (V)})}_(d) are compensated quantities,-   while {circumflex over (V)}_(q), {circumflex over (V)}_(d) are    derived from measurement.    For open winding detection, only the compensated voltage {circumflex    over ({circumflex over (V)})}_(q) is needed, which is equal to    {circumflex over (V)}_(q)+{circumflex over (V)}_(d)·K_(dωm). Hence,    this extra term “K_(d)·V_(dLN)·ω_(m)” is incorporated into the    equation for the voltage residue, r_(q), to compensate for filter    time delay.

Note τ is actually a relative time Δτ; if we measure ee using a low-passfilter with time lag τ_(o), and measure currents using a low-passfilter, also with time lag τ_(o), but V_(a), V_(b), and V_(c) are usingan LPF with time lag τ_(v), then we should use$K_{d} = {\frac{P}{2} \cdot \left( {\tau_{v} - \tau_{o}} \right)}$to calculate voltages that would correspond to the currents and phaseangles, which have time delay τ_(o) and not calculate the actualvoltages with no time delay.

1. A method for detecting an open winding condition in a motor, themethod comprising: a. measuring a winding voltage, a winding current anda motor speed; b. calculating a residue voltage for the winding, theresidue voltage equaling the difference between a measured voltage dropacross the winding and a calculated voltage drop for the winding, thevoltage drop calculated for a non-open winding condition; and c.comparing the residue voltage to a threshold value.
 2. A methodaccording to claim 1, further including: d. signaling when the residuevoltage exceeds the threshold value, to declare an open windingcondition.
 3. A method for detecting an open winding condition in adual-stator redundant motor, the method comprising: a. measuring a firststator winding voltage, a first stator winding current and the motorspeed; b. computing a first residue voltage for the first statorwinding, the first residue voltage equaling the difference between ameasured voltage drop across the first stator winding and a calculatedvoltage drop value for a non-open first stator winding; c. measuring asecond stator voltage across a second stator winding and a secondcurrent through the second stator winding; d. calculating a secondresidue voltage for the second stator winding, the second residuevoltage equaling the difference between a measured voltage drop acrossthe second stator winding and a calculated voltage drop value for anon-open second stator winding; e. calculating a residue voltagedifference equal to the magnitude of the difference between the firstresidue voltage and the second residue voltage; and f. comparing theresidue voltage difference to a threshold value.
 4. A method accordingto claim 3, the method further including: g. signaling when the residuevoltage difference exceeds the threshold value, to declare an openwinding condition.
 5. A method according to claim 4, the method furtherincluding: h. signaling when the first residue voltage exceeds a firstresidue threshold value to declare an open winding condition.
 6. Amethod according to claim 3, the method further including compensatingfor measurement delay before calculating a residue voltage difference.